The Bernoulli Equation Patrick Bruskiewich, M.Sc. Department of Physics and Astronomy, University of British Columbia 6224 Agricultural Road, Vancouver, B.C., Canada V6T 1Z1 A t » 4, Bernoulli's Principle: where the velocity of an incompressible fluid (gas or liquid) is high, the pressure is low and where the velocity is low, the pressure is high. Bernoulli's Principle makes sense: Fluids will flow from a region of higher pressure to a region of lower pressure and in doing so increase their speed as a result. If this were not the case => if the pressure were higher at a point of higher velocity then the fluid would be slowed down at that point. As a result this would be a point of lower velocity. This is a contradiction! ? Bernouilli developed an equation of state that expresses his principle quantitatively. Assume the fluid flow is steady and incompressible. fe£ " v Volume ~V, V^tavwe r V* f77777^' Consider the work to move a small volume V\ of fluid at point 1, W 1 = F x Ah = P 1 A 1 Al 1 (1) At point 2 the force exerted on the fluid is opposite to the motion so then W 2 = - F 2 Al 2 = - P 2 A 2 Al 2 (2) 3 Work is also done on the fluid to counter the force of gravity. The work done to lift the mass of fluid in volume Vi from point 1 to point 2 is W g = - mg(y 2 - y\) (3) So then the net work W net done on the fluid in moving it from point 1 to point 2 is and is given by W net = W 1 + W 2 + Wg = P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 -yi) (4) According to the work-energy principle, the net work on a system is equal to its change in kinetic energy. That is 1 1 W net = ARE — -mv 2 - -mvi (5) So then 1 9 1 o -rnv 2 - -mv 1 = P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 - Vi) (6) Gather the 1 and 2 terms together we find 1 2 P 2 A 2 Al 2 + -mv 2 + mgy 2 i t~\ PlA 1 Ali + - mv l + m 9V\ Z R We can now factor out the volume on each side of the equation, remembering that p = y t namely 1 2 A 2 AZ 2 (P 2 + ~pv 2 + P9V2) 1 2 = AiAZi(Pi + -pv 1 + P^yi) (7) We know from the continuity equation that A 2 AZ 2 = AiAZi, which is the volume of the fluid ele- ment flowing through the system. We can therefore cancel these terms off both sides of the equation. What we are left with is 1 2 P 2 + ~pv 2 + P9V2 Pi + ~pv{ + pgy ± This is Bernoulli's equation Since points 1 and 2 can be any points along a tube of flow, Bernouilli's equation can be writ- ten 1 2 p 2 + ~P v 2 + P 9 V2 = constant (8) a. at every point in the fluid where y is the height of the centre of the tube above the reference level. The Torrjcelli Apparatus b* %-y t v;*o %x -r r, AX H The top of the apparatus and the spigot are open to the atmosphere so that J\ = P 2 = *atm ■ Using Bernoulli's equation (> 2 « 0) we have ~P v i + P 9 Vi = P 9 V2 (9) Solving for v\ we find ,2 _ vf = 2g(v2 - vi) (io) so that vi = \][2g(y 2 - vi)] (11) This is known as Torricelli's theorem, which was discovered about 100 years before the Bernoulli equation was developed. Torricelli was a student of Galileo Galilei. Experiment using Torricelli's apparatus Objective: measure the distance the flow goes before hitting the ground (Ax) and relate this to the height of the fluid column (ft = y^ — 2/1). Data points for Ax and h are collected and a best fit power law curve is graphed. Assume that Ax and h are related by a power law given by V Ax = a ft] n (12) h where n is to be determined. By taking the natural logarithm of the data points and graphing the relationship we should find that the slope of the graph is the power of the relationship. That is U[$ X u[W A [In Ax] A [In ft] = n = slope (13) 10 Theoretical Prediction We know that Ax v = (14) At V J The distance m is related to the time it takes the drop to fall by 2/1 = §#t 2 , so then v = Ax 9 2j/i = \/[2 £ h] (15) where h = y 2 — 2/1 ■ This means that in theory Ax = 2[yi h]2 (16) where 2/1 is the height of the spigot above the ground. Theory therefore tells us that the slope of the datapoints on the log-log Ax and h graph should be equal to i 1 1